Theorem esumeq2d

Description: Equality deduction for extended sum. (Contributed by Thierry Arnoux, 21-Sep-2016)

	Ref	Expression
Hypotheses	esumeq2d.0	|- F/ k ph
	esumeq2d.1	|- ( ph -> A. k e. A B = C )
Assertion	esumeq2d	|- ( ph -> sum* k e. A B = sum* k e. A C )

Proof

Step	Hyp	Ref	Expression
1		esumeq2d.0	|- F/ k ph
2		esumeq2d.1	|- ( ph -> A. k e. A B = C )
3		eqidd	|- ( ph -> A = A )
4	2	r19.21bi	|- ( ( ph /\ k e. A ) -> B = C )
5	1 3 4	esumeq12dvaf	|- ( ph -> sum* k e. A B = sum* k e. A C )